Random Sturm Liouville Operators

Abstract

Selfadjoint Sturm-Liouville operators $H_{\omega }$ on $L_2(a,b)$ with random potentials are considered and it is proven, using positivity conditions, that for almost every $\omega$ the operator $H_{\omega }$ does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as $H_{\omega }$ but in $L_2(\tilde{a},\tilde{b})$ where $(\tilde{a},\tilde{b})\subset (a,b)$.